I had this on a recent exam and I just loved the solution so much, I thought I would share it. Enjoy.

means , the set of bounded linear operators on H? I don't think you need the separability of H.

Spoiler:

Since , by Riesz there exists a unique such that for all . So . Now since T is positive self-adjoint, we get a unique positive self-adjoint square-root , and . So the minimum value is , where is the adjoint to .

I had this on a recent exam and I just loved the solution so much, I thought I would share it. Enjoy.

means , the set of bounded linear operators on H? I don't think you need the separability of H.

Spoiler:

Since , by Riesz there exists a unique such that for all . So . Now since T is positive self-adjoint, we get a unique positive self-adjoint square-root , and . So the minimum value is , where is the adjoint to .

means compact operators, and though your solution works, there's a particularly cute one when you use that is compact.

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